Question

Matrices \(A\) and \(B\) are given. Solve the matrix equation \(A X=B\). $$ A=\left[\begin{array}{ccc} -3 & 3 & -2 \\ 1 & -3 & 2 \\ -1 & -1 & 2 \end{array}\right], \quad B=I_{3} $$

Short answer

The solution matrix \( X \) is \( \left[\begin{array}{ccc} 1 & 1 & 4 \\ 1 & 3 & -2 \\ 2 & -2 & 3 \end{array}\right] \).

Step-by-step solution

Understand the Problem

We are given a matrix equation \( A X = B \), where \( A \) and \( B \) are given matrices and we need to find the matrix \( X \). \( B \) is the identity matrix \( I_3 \) which implies we need the inverse of \( A \) such that \( X = A^{-1} B \).

Determine if Inverse Exists

Check if the inverse of \( A \) exists by calculating its determinant. The determinant of the matrix \( A \) is calculated as follows: \[\det(A) = -3(-6) - 3(2+2) - 2(1+3) = 18 - 12 - 8 = -2\] Since the determinant is not zero, \( A \) is invertible, and we can find \( A^{-1} \).

Find the Inverse of A

Use the formula for the inverse of a 3x3 matrix, which involves the adjugate matrix \( \,\text{adj}(A)\, \) and the determinant, \( \,\det(A)\, \): \( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \). The adjugate matrix is formed using minors, cofactors, and transposing the cofactor matrix of \( A \).

Calculate Adjugate of A

To find the adjugate, we calculate the matrix of cofactors and then transpose it. The cofactors are calculated by finding the determinant of the 2x2 submatrix obtained by deleting the current row and column. After calculating all cofactors, we obtain the adjugate \[ \text{adj}(A) = \left[\begin{array}{ccc} -2 & -2 & -8 \ -2 & -6 & 4 \ -4 & 4 & -6 \end{array}\right] \]

Compute Inverse Using Determinant

With \( \,\det(A) = -2 \, \) and the adjugate matrix found, compute the inverse: \[A^{-1} = \frac{1}{-2}\cdot \left[\begin{array}{ccc} -2 & -2 & -8 \ -2 & -6 & 4 \ -4 & 4 & -6 \end{array}\right] = \left[\begin{array}{ccc} 1 & 1 & 4 \ 1 & 3 & -2 \ 2 & -2 & 3 \end{array}\right] \]

Solve for X

Since \( B \) is the identity matrix \( I_3 \), the equation simplifies to \( AX = I \), meaning \( X = A^{-1} \). Based on the computed inverse, we find: \[ X = \left[\begin{array}{ccc} 1 & 1 & 4 \ 1 & 3 & -2 \ 2 & -2 & 3 \end{array}\right] \]

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It has many important properties, one of which is determining if a matrix is invertible. In simpler terms, for a 3x3 matrix like our matrix \( A \), the determinant helps us understand if we can "reverse" the matrix using its inverse.

To find the determinant of a 3x3 matrix, we use a specific formula that involves multiplying and adding elements from the matrix in a structured way. For our matrix \( A \), the determinant is calculated as follows:

• First, multiply the diagonals: \(-3(-6)\)
• Then subtract products from the other diagonals: \(-3(2+2)\) and \(-2(1+3)\)
• Add these results: \(18 - 12 - 8\)

The final result gives us \( -2 \), which tells us that our matrix \( A \) is invertible because the determinant is not zero.

Adjugate Matrix

An adjugate matrix is a key player when finding the inverse of a matrix. It is derived from the original matrix but uses a process called finding cofactors and then transposing them. This sounds complex, but let's break it down step-by-step.

First, we need to compute the cofactor of each element in the original matrix \( A \). A cofactor is essentially a minor (determinant of a smaller 2x2 matrix) with some added sign modifications based on its position. After computing all the cofactors, we rearrange them following a pattern known as transposing — flipping the rows and columns.

Finally, joining these revised elements forms the adjugate matrix \( \text{adj}(A) \). This matrix is necessary for computing the inverse, especially when used with the determinant of the original matrix.

Identity Matrix

The identity matrix is an incredibly handy tool in matrix mathematics. For a 3x3 matrix, the identity matrix, often denoted by \( I_3 \), looks something like this:

• \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \)

What makes the identity matrix special is its role as the "multiplicative identity" in matrix algebra. That means when any matrix is multiplied by the identity matrix, it remains unchanged.

When solving an equation like \( AX = B \), if \( B \) is the identity matrix, it indicates that \( X \) is essentially the inverse of \( A \). This happens because multiplying any matrix by its inverse results in the identity matrix. This property is extremely useful when solving systems of equations represented in matrix form.

3x3 Matrix Inverse

Finding the inverse of a 3x3 matrix involves a few straightforward steps, provided the determinant is non-zero. The inverse is a special matrix \( A^{-1} \) that, when multiplied by the original matrix \( A \), results in the identity matrix.

Start by calculating the determinant. If it's not zero, as in our example where \( \det(A) = -2 \), the matrix \( A \) can have an inverse. Next, determine the adjugate matrix using cofactors from \( A \), which involves some calculation and rearrangement.

Finally, apply the formula:

• \( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \)

This formula cleverly combines the determinant and the adjugate to produce the inverse matrix. And that gives us the toolbox we need to "reverse" the matrix operations and solve equations such as \( AX = B \), especially when \( B \) is the identity matrix (resulting in \( X = A^{-1} \)).